Becker’s conjecture on Mahler functions
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- by Jason P. Bell, Frédéric Chyzak, Michael Coons and Philippe Dumas PDF
- Trans. Amer. Math. Soc. 372 (2019), 3405-3423
Abstract:
In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker’s conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^\gamma Q(z)$ for some explicit nonnegative integer $\gamma$ and such that $1/Q(z)$ is $k$-regular.References
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Additional Information
- Jason P. Bell
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada
- MR Author ID: 632303
- Email: jpbell@uwaterloo.ca
- Frédéric Chyzak
- Affiliation: INRIA, Université Paris–Saclay, Paris, France
- Email: Frederic.Chyzak@inria.fr
- Michael Coons
- Affiliation: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales, Australia
- MR Author ID: 857151
- Email: Michael.Coons@newcastle.edu.au
- Philippe Dumas
- Affiliation: INRIA, Université Paris–Saclay, Paris, France
- MR Author ID: 342673
- Email: Philippe.Dumas@inria.fr
- Received by editor(s): April 20, 2018
- Received by editor(s) in revised form: November 23, 2018
- Published electronically: May 31, 2019
- Additional Notes: The research of the first author was partially supported by an NSERC Discovery Grant.
The third author was visiting the Alfréd Rényi Institute of the Hungarian Academy of Sciences during the time this research was undertaken; he thanks the Institute and its members for their kindness and support. - © Copyright 2019 by the authors
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3405-3423
- MSC (2010): Primary 11B85, 30B10; Secondary 68R15
- DOI: https://doi.org/10.1090/tran/7762
- MathSciNet review: 3988615